# Bivariate Regression

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Linear Regression Models
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SPSS for Windows® Intermediate & Advanced Applied Statistics Zayed University Office of Research SPSS for Windows® Workshop Series Presented by Dr. Maher Khelifa Associate Professor Department of Humanities and Social Sciences College of Arts and Sciences

© Dr. Maher Khelifa

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Bi-variate Linear Regression
(Simple Linear Regression)

© Dr. Maher Khelifa

Understanding Bivariate Linear Regression
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 Many statistical indices summarize information about particular

phenomena under study.

 For example, the Pearson (r) summarizes the magnitude of a linear

relationship between pairs of variables.

 However, one major scientific research objective is to “explain”,

“predict”, or
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The parameters β0 and β1 are constants describing the functional relationship in the population. The value of β1 identifies the change along the Y scale expected for every unit changed in fixed values of X (represents the slope or degree of steepness). The values of β0 identifies an adjustment constant due to scale differences in measuring X and Y (the intercept or the place on the Y axis through which the straight line passes. It is the value of Y when X = 0). ∑ (Epsilon) represents an error component for each individual. The portion of Y score that cannot be accounted for by its systematic relationship with values of X.

© Dr. Maher Khelifa

Understanding Bivariate Linear Regression
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The formula Y = β0 + β1X + ε can be thought of as:

Yi = Y’+ εi (where α + β1Xi define the predictable part of any Y score for fixed values of X. Y’ is considered the predicted score).

The mathematical equation for the sample general linear model is represented as:

Yi = b0 + b1Xi + ei.

In this equation the values of a and b can be thought of as values that maximize the explanatory power or predictive accuracy of X in relation to Y. In maximizing explanatory power or predictive accuracy these values minimize prediction error. If Y represents an individual’s score on the criterion variable and Y’ is the predicted score, then Y-Y’ = error score (e) or the